At the globe screen, you can press Y to bring up a list of your missions or scroll till you find one that has a "SOS" over the top of it. Select that mission to begin.
You must complete the mission in one life, if you die you will not be able to retry!
Continue until achievement pops.
To further help elaborate, here's what you must do for each type of SOS Mission:
Red SOS Missions mean you must complete the stage with someone else's Avatar.
Green SOS Missions mean you must complete the stage with either your Avatar or someone else's.
and Blue SOS Missions mean you must find and open a trap box on the stage before completing it. Completing the stage without doing this results in a Mission Failure.